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Limit Theory for an Explosive Autoregressive Process

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  • Xiaohu Wang

    ()
    (Chinese University of Hong Kong)

  • Jun Yu

    ()
    (Singapore Management University, School of Economics)

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Abstract

Large sample properties are studied for a rst-order autoregression (AR(1)) with a root greater than unity. It is shown that, contrary to the AR coe¢ cient, the least- squares (LS) estimator of the intercept and its t-statistic are asymptotically normal without requiring the Gaussian error distribution, and hence an invariance principle applies. While the invariance principle does not apply to the asymptotic distribution of the LS estimator of the AR coe¢ cient, we show explicitly how it depends on the initial condition and the intercept. Also established are the asymptotic independence between the LS estimators of the intercept and the AR coefficient and the asymptotic independence between their t-statistics. Asymptotic theory for explosive processes is compared to that for unit root AR(1) processes and stationary AR(1) processes. The coefficient based test and the t test have better power for testing the hypothesis of zero intercept in the explosive process than in the stationary process.

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Bibliographic Info

Paper provided by Singapore Management University, School of Economics in its series Working Papers with number 08-2013.

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Length: 19 pages
Date of creation: Nov 2013
Date of revision:
Publication status: Published in SMU Economics and Statistics Working Paper Series
Handle: RePEc:siu:wpaper:08-2013

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  1. Bent Nielsen, 2005. "Analysis of co-explosive processes," Economics Papers 2005-W08, Economics Group, Nuffield College, University of Oxford.
  2. Tom Engsted & Bent Nielsen, 2010. "Testing for rational bubbles in a co-explosive vector autoregression," CREATES Research Papers 2010-25, School of Economics and Management, University of Aarhus.
  3. Peter C.B. Phillips, 1985. "Time Series Regression with a Unit Root," Cowles Foundation Discussion Papers 740R, Cowles Foundation for Research in Economics, Yale University, revised Feb 1986.
  4. Peter C.B. PHILIPS & Yangru WU & Jun YU, 2009. "Explosive Behavior in the 1990s Nasdaq: When Did Exuberance Escalate Asset Values?," Working Papers 19-2009, Singapore Management University, School of Economics.
  5. Peter C.B. Phillips & Tassos Magdalinos, 2004. "Limit Theory for Moderate Deviations from a Unit Root," Cowles Foundation Discussion Papers 1471, Cowles Foundation for Research in Economics, Yale University.
  6. Peter C.B. Phillips & Tassos Magdalinos, 2007. "Limit Theory for Explosively Cointegrated Systems," Cowles Foundation Discussion Papers 1614, Cowles Foundation for Research in Economics, Yale University.
  7. Magdalinos, Tassos, 2012. "Mildly explosive autoregression under weak and strong dependence," Journal of Econometrics, Elsevier, vol. 169(2), pages 179-187.
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